If `0^(@)lethetale90^(@)` , then solve the following equations :
(i) `(costheta)/(1-sintheta)+(costheta)/(1+sintheta)=4`
(ii) `(cos^(2)theta-3costheta+2)/(sin^(2)theta)=1`
(iii) `(costheta)/("cosec"theta+1)+(costheta)/("cosec"theta-1)=2`

Answer

Let's solve the given equations step by step.

### Given:
1. \( \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4 \)
2. \( \frac{\cos^2 \theta - 3 \cos \theta + 2}{\sin^2 \theta} = 1 \)
3. \( \frac{\cos \theta}{\csc \theta + 1} + \frac{\cos \theta}{\csc \theta - 1} = 2 \)

### Solution:

#### (i) Solve \( \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4 \)

1. **Combine the fractions:**
\[ \frac{\cos \theta (1 + \sin \theta) + \cos \theta (1 - \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} = 4 \]
This simplifies to:
\[ \frac{\cos \theta (1 + \sin \theta + 1 - \sin \theta)}{1 - \sin^2 \theta} = 4 \]
\[ \frac{2 \cos \theta}{\cos^2 \theta} = 4 \]

2. **Simplify the equation:**
\[ \frac{2}{\cos \theta} = 4 \]
\[ \cos \theta = \frac{1}{2} \]

3. **Find \( \theta \):**
\[ \theta = \cos^{-1} \left(\frac{1}{2}\right) = 60^\circ \]

#### (ii) Solve \( \frac{\cos^2 \theta - 3 \cos \theta + 2}{\sin^2 \theta} = 1 \)

1. **Multiply both sides by \( \sin^2 \theta \):**
\[ \cos^2 \theta - 3 \cos \theta + 2 = \sin^2 \theta \]

2. **Use the identity \( \sin^2 \theta = 1 - \cos^2 \theta \):**
\[ \cos^2 \theta - 3 \cos \theta + 2 = 1 - \cos^2 \theta \]
\[ 2 \cos^2 \theta - 3 \cos \theta + 1 = 0 \]

3. **Factor the quadratic equation:**
\[ (2 \cos \theta - 1)(\cos \theta - 1) = 0 \]

4. **Solve for \( \cos \theta \):**
\[ 2 \cos \theta - 1 = 0 \quad \Rightarrow \quad \cos \theta = \frac{1}{2} \quad \Rightarrow \quad \theta = 60^\circ \]
\[ \cos \theta - 1 = 0 \quad \Rightarrow \quad \cos \theta = 1 \quad \Rightarrow \quad \theta = 0^\circ \]

#### (iii) Solve \( \frac{\cos \theta}{\csc \theta + 1} + \frac{\cos \theta}{\csc \theta - 1} = 2 \)

1. **Convert \( \csc \theta \) to \( \frac{1}{\sin \theta} \):**
\[ \frac{\cos \theta}{\frac{1}{\sin \theta} + 1} + \frac{\cos \theta}{\frac{1}{\sin \theta} - 1} = 2 \]

2. **Combine the fractions:**
\[ \frac{\cos \theta \sin \theta}{1 + \sin \theta} + \frac{\cos \theta \sin \theta}{1 - \sin \theta} = 2 \]
\[ \frac{\cos \theta \sin \theta (1 - \sin \theta) + \cos \theta \sin \theta (1 + \sin \theta)}{(1 - \sin \theta)(1 + \sin \theta)} = 2 \]
\[ \frac{2 \cos \theta \sin \theta}{\cos^2 \theta} = 2 \]

3. **Simplify the equation:**
\[ \frac{2 \sin \theta}{\cos \theta} = 2 \]
\[ \tan \theta = 1 \]

4. **Find \( \theta \):**
\[ \theta = 45^\circ \]

### Summary of Solutions:
- From (i): \( \theta = 60^\circ \)
- From (ii): \( \theta = 0^\circ \) or \( 60^\circ \)
- From (iii): \( \theta = 45^\circ \)